Synchronization of finite-dimensional “force” generators

654

L. M. Markhashov

BIBLIOGRAPHY

1. 2.

3. 4. 5. 6. 7. 8. 9.

Polak, L. S., Variational Principles of Mechanics. M., FizmatgiG 1960. Noether. E., Invariante Transformationsproblem. Nachr. Kiinigl. Gesellsch. d. Wiss. zu Gotingen, math. -phis.Kl., Abs.2, 1918. Sdkoli k, G. A. , Group Theory Methods in the Theory of Elementary Particles. 1965. M. t Atomizdat, Ovsiannikov. L. V., Groups and group invariant solutions of differential equations. Dokl. Akad.Nauk SSSR. Vol. 118, No3, 1958. Ovsiannikov, L. V., Group Classification of Differential Equations. Novosibirsk, 1zd.S. 0. Akad. Nauk SSSR, 1962. Klein, F., Comparative review of the latest investigations in geometry (Erlangen program). Coll. “Fundamentals of Geometry”. M., Gostekhizdat, 1956. Chebotarev, N. G., Theory of Lie Groups. M. -L., Gostekhteorizdat, 1938. Theorie der Transformationsgruppen, Leipzig, Teubner. Abs. 1. 1888. Lie.S., Finikov, S. P., Method of External Cartan Forms in the Differential Geometry. M.-L., Gcstekhizdat, 1948. Translated by L. K.

SYNCHRONIZATIONOF FINITE-DIMENSIONAL "FORCE" GENERATORS PMM Vol. 32, Np5, 1968, pp. 825-833 R, F. NAGAEV (Leningrad) (Received June 5, 196 8) The problem of synchronizing almost conservative dynamic objects [l] under weak constraints is considered. The character and mechanism of action of the objects on the sup.porting system are defined qualitatively and are no way related to its specific form 121, The proposed procedure for investigating synchronous states in such systems is based on the notion of the dynamic influence matrix. It is shown that qualitative definition (specification) of the character of action of the objects is a natural basis for their classification. The paper ends with an examination of the synchronization of generators of “forces” of the simplest type, i.e. of one- and two-dimensional “forces”. The results can be applied, for example, to-the solution of vibration engineering problems involving the properties of several complex vibration sources operating simultaneously. The problems of synchronous state stability have already been investigated in [3 and 41, and therefore will not be considered here. Let us assume that the motion of the 1. The dynamic influence matrix. arbitrary i th object (i = 1.. . , n) in a system is completely defined if we know the time variation of li X 1 vector columns of its proper coordinates Qi = (qi(t), . . . . qi(‘i)) and mi X 1 vectors of the reverse influence parameters x1 = (xi(t), . . . , xicmiJ). The physical character of the reverse influence parameters is completely determined by the specifies of the object and is unrelated to the form of the supporting system @I. The

Synchronization of fir&e-dimcnslmal

“force” generators

855

additional kinetic and additional potential energy of the object [lf can be naturally ten in a form invariant relative to the form of the supporting system. ATi = xi*‘A,,

writ-

(Xi 3 qi) qi’ + ‘12X.iY’A, (Xi 9 qi) Xi’ AITi z AlJi’(xi,

(1:l)

qi)

pi X li and rni X rrzi matrices with not small Here A,, and A, are, respectively, On the coefficients dependent on xi and qi ; the asterisk denotes matrix transposition. other hand. the time variation of the reverse influence parameters is determined exclusively by the variation of the components of m X i vectors y = (ran, , , . , y,J of the absolute coordinates of the system.so that x’i - Di (y) (1.2) We note that the choices of the vectors y and si’(i = 1,... . . . . ~21are in general completely unrelated. In practice the vector of reverse influence parameters often has a clear geometric significance, while the components of the vector y are sometimes fairly abstract (e. g. in the case of a supporting system with distributed parameters, when nb = 00). Since the oscillations where p, is the coupling

of the supporting parameter,

xi = pui + j.C . - .

system are small [I]. so that =/.Lv Y we have the expansion

(I+ = Dl,,$,

Dim = dlli

(I.31 (1 *a)

( y) f dy frz0

Here Di, is an rnf X m matrix with constant coefficients. Relations (1.1) with allowance for (1.3) can be rewritten as ATi =

p~i*‘Amf (0, qi) qi’+

pa we *)

AIIi mypUi*Ci (qi) $- pz

so that the additional kinetic potential of the i th object to within quantities order of smallness is PAT,, = [L {Ui*‘A,i((‘r qi) q*’ - Ui*C, (qi)]

l

**

(f-5)

of a higher (1.6)

The potential generalized force conveyed to the supporting system by the ith object is given with the same degree of accuracy by

Here

Dmi = Dim*

-&$+ALi= C and the nfX 1

-

D,iF,

vector

Fizz--

$-&-$ ALi= (‘1-7) - & [4ni to9 qi) q;l i- ci (Cri) i i) ( is the force due to the irh object reduced to its proper reverse influence parameters, i.e. the vector of fixed actions of the object on the supporting system. Thus, F, is the characterizes the physical “force” generated by the i th object, and the matrix I),, distribution of actions of this force over the supporting system. If the vector of the reverse influence parameters has three orthogonal spatial components, then the vector Pi is a force in the ordinary mechanical sense of the word ; but if the reverse influence parameters are rotations, then I;‘, varies periodically with time and is a function of the proper rapidly rotating

F, = Ft ($_+ Cli, V)

phase.

(1s= vt)

0 J3) where ai is the phase shift of the synchronous motion of the i th object. The differential equation of small oscillations of the supporting system in the generan ting approximation is &IV”

+

Bv’ + CV =

2 D,iF, $zzL

(Z + Ui, V)

(1.0)

856

II.F. Nagaev

where hf and c are symmetric positively defined m X m matrices with constant coefficients. The symme~ic part of the m X m matrix B is also positive. Let us introduce the impulsive periodic function (11 (T) representable as a generalized Fourier series,

(1.10) The periodic solution of system (1.1) can be represented

as a contraction

[SJ.

where the m X m matrix Iv,,, is the pulse-frequency characteristic of the supporting system which satisfies Eq. (1.12) va&l’I”,” + vB’~~,’ -+ C’Ym= E,(I) (t) Here 15, is a unit m X m matrix, and the prime denotes differentiation with respect to r. The vector of reverse influence parameters is given (in accordance with (1.4)) by Formula

(1.13)

Here the mi X rnjmatrix

__

Kij = DimY,Dmj

(l.lr,)

is the dynamic matrix of influence of an arbitrary j th object on the 2 th object. Now let us assume that the system of forces exerted by the objects is of the form l?j = l.2~~~1’(T)-6[j

(1.15)

where I is a fixed natural number (1 < 2 6 22). Then, by virtue of (1.13). we have U{“) = Iii1 (tc Y)

(1*1(i)

Thus, the component Kij(P’ 9) (p = 1,. ,. , m,; q = I,,. , , ITZj) of the matrix I$f yields the law of variation of the p th reverse influence parameter of the i th object on the periodic impulsive perturbation ‘f)(t) from the q th output of the i th object. If small oscillations of the sup~rt~g system are not accompanied by the action of characteristic w, gyroscopic forces, so that 1% = IS*, then the matrix pulse-frequency of the supporting system is also symmetric. Because of this we have the conditions of dynamic reciprocity (1.27) Iiij = ICji* Finally, we note that by (1.16)

are matrix harmonic influence tem, then, of course, K{$ =

the Fourier coefficients

coefficients.

of the expansion

If there is no friction

of the dvnamic

in the supporting sys-

0.

2. The condition8 of sxfrtencs of a synchronous mode of a sy 8 t 6 m , It is shown in [P] that the generating config~ation of phase shifts of synchronous motions of almost conservative objects under supporting constraints can be determined from the system

(2.1)

Here ICi (v) is the slop

of the skeleton curve of the jth object, and fr (v)

is the

857

sycabmniaaUon of fix&e-dimmnianal “force” g-&toss

motion of its nonpotential forces in the generating approximadon averaged over the period of synchronous motion. The partial vibrational moments w,j can be determined from Formulas W,j

=

2(1>,j

II

~

(~

-

~) (2.21

The possibility of introducing the quantities AA,, and ‘I),, is due to the fact that small oscillations of the supporting system can be represent&d as sum (1.12), each of ¶n whose terms (2.3) vi (% + ui, Y) L-- VI, (z - r(, Y) D,,,iPi (9 + ait v) d”I s 0

characterizes the con~bution of the corresponding object, The equations for determlng the components of can be written as

Mv('+ PVC+ Cvi = where AL,, is the additional kinetic potential the 1 th object in the generating approximation. A&j = vj*‘DrmAmr

(2.4)

of the i th object due to the motion of

(0, qr) ql’ -- vj*DimG

(qt)

The energy characteristics of interaction of the i th and i th objects averaged the period of synchronous motion, A A,, and CT),, , are given by

(2.5) over

a*

A&j (ai-aj,

V) =

&

5

ALi; dr

0 2%

@fj(@-f-

aj, V) = &

.s

+

vi*‘R,vi’ dr

0

It is not difficult to show that basic equations (2.1) remain unaltered in our more general case of objects with an arbitrary number of degrees of freedom. Let us take the scalar product of matrix equation (2.4) and the vector row rj**and add the result to the analogous expression with the subscripts i and f interchanged. Let us then average the result over the period of synchronous motion. Then, after some transformations we arrive directly at the relations

(2.7) We can show that Formula (2.7) remains valid for i =i, if we assume that the quantities qi’ and qt in the expression for ALii (2.5) are independent of the phase shifu c+. By virtue of this assumption we have and the initial equations can be rewritten as

(2.1)

for determining

the generating

phase shift configuration

It is natural to call the quantity A:lI in this expression the “addftional action integral of the i th object”. This is the general formulation of the conditions of existence of a synchronous mode which does not include the energy characteristics of motion of the supporting system. Let us now turn to the determination of the additional action Integral of the ith object which can be obtained by averaging Expression (1.8). By some simple transformations

R. F. Nagaev

858

involving

integration

by parts, allowing

for (1.9) and (1.4), we obtain

Ahi=~,~~~Fi*(t’Zi,V)f;~j(T-q,y)~j(’~~~j,”)drdli J=l,I 0 By virtue of (1.12). relation

(2.10) can be rewritten

(2.1U) as

2%

(2.11)

AAi = & 5 Fi* (T + ai, Y) Iii (~3 Y) do 0

Thus, the additional action integral of the i th object is equal to the generalized force generated by the object(over its reactive vector displacement over the vector of reverse influence parameters) averaged over the period. An expression for the partial vibrational moments (2.2) can be obtained by differentiating Expression (2.10) with respect to the phase shift CL~in accordance with (2.9). By virtue of assumption (2.8) we must assume here that the vectorial force Fi (q ?r- Olir v) in the i th term of sum (2.10) is independent of ai. Hence, integrating by parts the derivative of Expression (2.10) with respect to ai, we obtain

(2.12)

2x 2s M’ij

(a, -

aj9

v) =

ss

7&

0 n

Fi* (r +

Ui,

v) Kij’ (V -

‘1,

Y) Fj (q + aj, V) dr dil

We note that in the absence of friction in the supporting system dynamic influence matrix (1.17) is an even function of time, Kij(t, V) = Kij( - T, Y), so that

Kij’ (to Y) = - Kij’ (-

T, Y).

Recalling dynamic reciprocity conditions (1.16), we readily infer from this that the matrix of partial vibrational moments is skew-symmetric,

Wij(Cti - CYzjr Y) $ Wji (0ij - CCirY) =O The above relation is at the same time a consequence chronous motion stability [l].

of the integral

(2.13) criterion

of syn-

3. Synchronization of generator, of force8 of the timplsrt ty pe , The process of constructing a system of equations for determining the generating phase shift configuration can be broken down into three steps: 1) determination of the magnitudes and directions of the forces l”; exerted by the objects with a braked supporting system ; 2) determination of the components of the dynamic influence matrix ; 3) execution of simple matrix operations and integrations in accordance with Expression (2.12). This enables us to classify synchronized objects according to the character and magnitude of the forces they generate. The most simple and at the same time one of the most common classes of objects of this type is clearly the class of generators of scalar influences, or less precisely, generators of forces of constant direction. In this case the reverse influence parameter vector has only one component. For this reason all of the quantities appearing in relations (2.12) are scalar. The harmonic coefficients in Expansion (1.19) are also scalar. Let “ij”? = );iF) sin 9:;) A+“) = k!!) Cos I$?’ (3.1) (1 II II where kij’F’ and \I, ii(E) are the influence coefficients of a unit harmonic force (in the generalized sense) of frequency s in dimensionless time acting at the output of the J’th

860

rotation

R. F. Nagaev

of the two vibrators is synchronized

in some way, so that the sum inertial

force

v) = 2 me+ cos (‘F + a)

fl(x+a,

has a constant direction when the supporting system is braked. The distance between the centers of rotation of the vibrators is small, and the point 0 is attached directly to the supporting system. Another example of the force generator of the same category is the crankgear mechanism (Fig.2). We assume that the crank is balanced and has a large moment of inertia, that the connecting rod is weightless, that the axis of rotation and the generatirices are connected to the supporting system, and that the ratio e / 1 is of the order of smallness of the coupling parameter. The force generated by this mechanism has the constant direction 0% in the generating approximation ; it is conveyed to the supporting system through the point 0 and is given by P (‘c + a, v) = meV%os (r + a) 2. A tor of the described debalances generated

mechanical harmonic moment generator. The simplest generaharmonic moment of constant direction is the coupled mechanical vibrator at the beginning of the preceding Section if the rotation of the constituent is synchronized and cophased as shown in Fig. 3. The mechanical moment by the vibrator is perpendicular to the plane of the figure and is given by F (z + a, v) = 2mervscos (7 + a)

Now let us consider the class of two-dimensional force generators one step more complicated than those described above. Limiting ourselves to purely harmonic perturbations

Fig. 4

Fig. 3

by the objects, we choose their proper coordinates in such a way that the force at the first output is maximum at the instant when the rotation phase vanishes. Here, of course, we have F!‘) COS(t+ a*) I= . 1),,. ,n (3.8) Fi (t + ai, v) = * F(a) co.9 (t + a. yi) 1 t I The dynamic influence matrix of the jth object on the i th, or, more precisely, its purely harmonic part, can be written as k$.l) Iiij

(t,

V) zz + I

The coefficients (in the mechanical

k;$”

cos

(Z-i

~0s (7 -

*p’)

j&2) 13

CQS (7 -

qp,

tft$“)

kg.2)

c
~>fj’2’)

(XCI) /

klj (P,@ and ~&~(P,q)(p, Q = 1, 2) appearing in matrix (3.2) signify case) the harmonic coefficients of the influence of the force (or

Syncbrcmieetfao

of firrite-dimensiatal“face” generators

861

moment, depending on the physical nature of the action) on the amplitude and phase of the displacement (or rotation). Substitute of Expressions (3.8) and (3.9) into basic relation (2.12) yields the following general formula for the partial vibrational moment: (3.10) Wij =T$1 [Pi” Fp$‘)

sin (o”$-

aj + lQ”1) + R~“lF~)k$2) sin (ai - aj + rj + (p~~~*~) +

+ ~~*)~~~)~~~l) sin (ai - ai - Tt + *$11) f- F(s) ~~z)~~?.~) sin (a *

I

II

f i

-aj-Tr

r.

+ J

qp’)] 11

The most common object of this class is a rotating force generator, i. e. an ordinary mechanical debalance vibrator (Fig. 4). The rotating vectorial force generated by a debalance generator is given by cos(r+aJ F(f+a, v)=me3 (3.111 sin (r + a) I Specialization of Expression (3.10) in accordance with this formula yields the expression for the partial vibrational moment of the problem of synchronizing mechanical vibrators mounted on a linear (more precisely, a linearizable) supporting system (‘I]. Some practical problems in vibration proofing involve the synchronization of generators of a helical “force” (dynamic screw). We are referring to a two-output dynamic object which generates a mechanical force and moment having the same constant direction with a braked supporting system.

Fig. 5 It is clear that one of the compouents of the vector of reverse influence parameters of a dynamic screw generator is linear displacement along some axis, while the other component is rotation about the same axis. A dynamic screw generator can take the form of a coupled vibrator whose component flux rotations are synchronized and cophased as shown in Fig. 5. The object generates the helical “force” F (z.+ a + v) = Zmeva

cos (r + a)

(3.12)

which is conveyed to the supporting system through the point 0 . For fl = 112 n this object degenerates into a harmonic force generator; for 8 = 0 it degenerates into a moment generator. We note in conclusion that the order of conversion of the components of the reverse influence parameter vector, and sometimes (e.g. in the case of a rotating force generator) the very choice of these components in problems involving synchronization of multidimensional force generators, are somewhat arbitrary and are generally dictated by purely physical considerations.

862

R. F. Nagaev

BIBLIOGRAPHY 1. 2. 3. 4. 5.

6.

7.

Nagaev, R. F., General problem of synchronization in an almost conservative system. PMM Vol.29, No5. 1965. Khodzhaev. K. Sh., On the excitation of vibrations. In&r. Zh. Mekh. Tverdogo Tela, Nn2, 1968. Synchronization in a system of essentially nonlinear objects with Nagaev, R. F., a single degree of freedom. PMM Vol.29, Nn2, 1965. Khodz haev, K. Sh., Synchronous motions in a system Nagaev. R. F. and of objects with supporting constraints. PMM Vol. 31, No4, 1967. Applying integral equations in the theory of nonlinear Rosenvasser, E. N., oscillations. Proceedings of the Second All-Union Congress on Theoretical and Applied Mechanics, Vol. 2, “Nauka” Press. Moscow. 1965. Nagaev, R. F, and Popova. I. A., Self-synchronization of several mechanical vibrators mounted on a single beam-type operating element. Inzh. Zh. Mekh. ‘Tverdogo Tela, N’ 1, 1967. Synchronization of mechanical vibrators coupled with a Khodzhaev. K. Sh., linear vibrational system. Inzh. Zh. Mekh. Tverdogo Tela. Np4, 1967. Translated

by A.Y.

STATISTICALMECHANICS OF GASEOUS SUSPENSIONS. DYNAMIC AND SPECTRAL EQUATIONS PMM Vol. 32, N’5, 1968, pp. 834-843 Iu. A. BUEVICH (Moscow) (Received July 25, 1968) We present the model of random, pseudoturbulent phase motions in concentrated, disperse, gas-suspended particles systems, based on the development of ideas expounded in our previous papers [l- 31 and [4]. This model enables us, in principle, to construct a structural theory of gaseous suspensions when the flow is pseudoturbulent [S], to compute the corresponding transfer coefficients and to formulate the dynamic equations of motion. Papers [l - 33 considered pulsating motions of a two-phase disperse system using the statistical approach and developed a general method of the quantitative treatment of the pulsations and their influence on the average motion of the system. At the same time, ways were indicated towards constructing a non-Newtonian mechanics of disperse systems. The model [3] however, retains a number of unsolved difficulties. First of them concerns the fact that the proposed model is based on the use of certain random forces acting on the phases in random motion, and of the statistical white noise, the latter allowing the description of not only of the orderly degeneration of the fluctuations of the averaged hydrodynamic field of a disperse system, but also of their random accumulation. The forces and the white noise enter 133 separately, although the general physical considerations imply that a mere appearance of the white noise should be the result of the action of the random forces.